Page 158 - Computational Colour Science Using MATLAB

CHARACTERIZATION OF HALF-TONE PRINTERS 145

9.4 Characterization of half-tone printers

9.4.1 Correction for non-linearity
If we consider a single ink printed on a substrate in a half-tone pattern and
denote the reflectance of the unprinted substrate by P and the reflectance of
w
the solid ink by P , then the Murray–Davies relationship (Yule, 1967) predicts
s
the measured reflectance P of the print. The value of P is related to the sum of the
reflectances of the two components weighted by their fractional area coverage,
P ¼ AP s þð1 AÞP w , ð9:1Þ

where A is the proportional area of the paper that is covered by ink.
Equation (9.1) can be inverted to predict the proportional dot area,

A ¼ðP w PÞ=ðP w P s Þ. ð9:2Þ
The simple Murray–Davies equation does not take dot gain into account. Yule
and Nielsen (1951) proposed a correction to the Murray–Davies equation,
1=2 1=2 2
P ¼½AðP s Þ þð1 AÞðP w Þ Š . ð9:3Þ
Equation (9.3) results in a non-linear relationship between the area coverage A
and the resulting reflectance P. The generalized Yule–Nielsen equation allows an
exponent n so that
1=n 1=n n
P ¼½AðP s Þ þð1 AÞðP w Þ Š , ð9:4Þ
where n usually is given a value between 1.0 (for a glossy substrate) and 2.0 (for a
matt substrate). This non-linear relationship is required to account for the
phenomenon of optical dot gain. Optical dot gain is the phenomenon that half-
tone prints usually appear darker than expected [based on Equation (9.1)]
because some light that strikes the unprinted substrate is absorbed by the ink
dots. This occurs because of light scattering in the substrate (Figure 9.2).
In addition to optical dot gain it is also necessary to consider mechanical dot
gain, which is the phenomenon where the printed dots usually are physically
larger than their target sizes because of flow of the wet ink when it is applied to
the substrate. The effect of mechanical dot gain is that a non-linear relationship
exists between the digital input count d and the dot coverage A. For given values
of d and P, the optimum area coverage A may be computed using
X X
1=n 1=n 1=n 1=n Þ= 1=n 1=n
A ¼ ðP s ðlÞ PðlÞ ÞðP s ðlÞ P w ðlÞ ðP s ðlÞ P w ðlÞ Þ,
ð9:5Þ
where it is assumed that there are no inter-colorant interactions (Bala, 2003).
Thus, P is measured for a number of levels d and then Equation (9.5) is used to
j
j
determine A . This procedure yields pairs of [d A ] from which a continuous
j
j
j
function can be derived that maps the digital count d to dot area coverage A.